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Generally speaking, all the first three just mean small river, but I think run is relatively uncommon/dialectal. Probably just a personal interpretation, but I tend to think of a creek as a small watercourse that over time has carved a lower bed than a brook/stream. Not that the water itself is deep - just that its surface is often significantly lower than surrounding ground level.
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FumbleFingersJul 4 '12 at 17:42

1 Answer
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Generally, the difference is size: you can step over a brook, jump over a creek, wade across a stream, and swim across a river. But the distinction between them (especially creek and stream) is somewhat hazy, and depends on who named them and when they were named. A run (such as Bull Run in Virginia) is a "small stream".

I agree with the ranking in that previous answer, but not with the criteria, which are not conclusive because many exceptions exist.